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The specific case of the inner product in Euclidean space, the dot product gives the product of the magnitude of two vectors and the cosine of the angle between them. Along with the cross product, the dot product is one of the fundamental operations on Euclidean vectors.

Since the dot product is an operation on two vectors that returns a scalar value, the dot product is also known as the scalar product.

Geometrically, one can also interpret the dot product as
\[ \vec{a} \cdot \vec{b} = \left( |\vec{a}| \right)\left( |\vec{b}| \cos{\theta} \right). \]
That is, one can view the dot product as the magnitude of \( \vec{a} \) times the magnitude of the component of \( \vec{b} \) that points along \( \vec{a} \). \(\left( |\vec{b}| \cos{\theta} \right) \) is the magnitude of the projection of \( \vec{b} \) onto \( \vec{a} \).

Similarly,
\[ \vec{a} \cdot \vec{b} = \left( |\vec{b}| \right)\left( |\vec{a}| \cos{\theta} \right), \]
so the dot product can also be viewed the magnitude of \( \vec{b} \) times the magnitude of the component of \( \vec{a} \) that points along \( \vec{b} \).

Since \(\left\|\vec{a}\right\|\) and \( \left\|\vec{b}\right\|\) are positive quantities, the sign of the dot product depends on \( \theta \):

If \(\ \theta \) is acute, then \( \cos{\theta} \) is positive, and therefore the dot product is positive.

If \(\ \theta \) is \(90^{\circ}\), then the dot product is zero. Vectors whose dot product vanishes are said to be orthogonal.

Given that the magnitude of \(\vec{a}\) is \(7\) and that of \(\vec{b}\) is \(8\), find \(\vec{a} \cdot \vec{b}\) when the angle between \(\vec{a}\) and \(\vec{b}\) is \[\]
\[\begin{array} &\text{(i)}~ 60^\circ &&&\text{(ii)}~ 90^\circ &&&\text{(iii)}~ 120^\circ. \end{array}\]

We know \(\left\|\vec{c}\right\| = 4\) and \(\left\|\vec{d}\right\| = 2\). Also, the two vectors are parallel, so \(\theta = 0\) and therefore \(\cos \theta = 1.\)
After we substitute the values in the formula, we get

\[\vec{c} \cdot \vec{d} = 4 \times 2 \times 1 = 8. \ _\square\]

True or False?

If \( \vec{a} \) and \( \vec{b} \) are vectors such that \( \vec{a} \cdot \vec{b} = 1 \),
then \( \vec{a} \) and \( \vec{b} \) must be parallel.

True
False

Properties

The dot product has several important and useful properties. Their proofs are fairly straightforward and left as exercises for the reader.

In other words, the product of two vectors in Cartesian coordinates is simply sum of the product of each of the corresponding components of the two vectors. The same applies to vectors in more than two dimensions.